Solve each of the following system of inequations in R.5x – 7 < 3(x

1.	Solve each of the following system of inequations in R.5x – 7 < 3(x + 3), 1 - 3x/2 ≥ x - 4
Answer» Given, 5x – 7 < 3(x + 3) and 1 - 3x/2 ≥ x - 4 Let us consider the first inequality. 5x – 7 < 3(x + 3) ⇒ 5x – 7 < 3x + 9 ⇒ 5x – 7 + 7 < 3x + 9 + 7 ⇒ 5x < 3x + 16 ⇒ 5x – 3x < 3x + 16 – 3x ⇒ 2x < 16 ⇒ \(\frac{2x}{2}\) < \(\frac{16}{2}\) ⇒ x < 8 ∴ x ∈ (–∞, 8) ...(1) Now, Let us consider the second inequality. 1 - \(\frac{3x}{2}\) ≥ x - 4 ⇒ \(\frac{2-3x}{2}\) ≥ x - 4 ⇒ \((\frac{2-3x}{2})\) \(\times\) 2 ≥ (x - 4) \(\times\) 2 ⇒ 2 – 3x ≥ 2(x – 4) ⇒ 2 – 3x ≥ 2x – 8 ⇒ 2 – 3x – 2 ≥ 2x – 8 – 2 ⇒ –3x ≥ 2x – 10 ⇒ 2x – 10 ≤ –3x ⇒ 2x – 10 + 10 ≤ –3x + 10 ⇒ 2x ≤ –3x + 10 ⇒ 2x + 3x ≤ –6x + 10 + 6x ⇒ 5x ≤ 10 ⇒ \(\frac{5x}{5}\) ≤ \(\frac{10}{5}\) ⇒ x ≤ 2 ∴ x ∈ (–∞, 2] ....(2) From (1) and (2), we get x ∈ (–∞, 8) ∩ (–∞, 2] ∴ x ∈ (–∞, 2] Thus, The solution of the given system of inequations is (–∞, 2].

Solve each of the following system of inequations in R.5x – 7 < 3(x + 3), 1 - 3x/2 ≥ x - 4

Answer»

Given,

5x – 7 < 3(x + 3) and 1 - 3x/2 ≥ x - 4

Let us consider the first inequality.

5x – 7 < 3(x + 3)

⇒ 5x – 7 < 3x + 9

⇒ 5x – 7 + 7 < 3x + 9 + 7

⇒ 5x < 3x + 16

⇒ 5x – 3x < 3x + 16 – 3x

⇒ 2x < 16

⇒ \(\frac{2x}{2}\) < \(\frac{16}{2}\)

⇒ x < 8

∴ x ∈ (–∞, 8) ...(1)

Now,

Let us consider the second inequality.

1 - \(\frac{3x}{2}\) ≥ x - 4

⇒ \(\frac{2-3x}{2}\) ≥ x - 4

⇒ \((\frac{2-3x}{2})\) \(\times\) 2 ≥ (x - 4) \(\times\) 2

⇒ 2 – 3x ≥ 2(x – 4)

⇒ 2 – 3x ≥ 2x – 8

⇒ 2 – 3x – 2 ≥ 2x – 8 – 2

⇒ –3x ≥ 2x – 10

⇒ 2x – 10 ≤ –3x

⇒ 2x – 10 + 10 ≤ –3x + 10

⇒ 2x ≤ –3x + 10

⇒ 2x + 3x ≤ –6x + 10 + 6x

⇒ 5x ≤ 10

⇒ \(\frac{5x}{5}\) ≤ \(\frac{10}{5}\)

⇒ x ≤ 2

∴ x ∈ (–∞, 2] ....(2)

From (1) and (2), we get

x ∈ (–∞, 8) ∩ (–∞, 2]

∴ x ∈ (–∞, 2]

Thus,

The solution of the given system of inequations is (–∞, 2].